MFG Series

Section 9.3 Series

Subsection Introduction

Often we are more interested in the sum of a sequence than in the sequence itself. For example, suppose that for 5 years you have made payments of $100 a month into a savings account that pays 12% annual interest compounded monthly. The sequence

\begin{equation*} p_n = 100(1.01)^n \end{equation*}

gives the current value of the payment you made $n$ months ago (plus interest). However, you are probably more interested in the total amount in your savings account, which is the sum of the terms $p_n$ from $n=1$ to $n=60\text{.}$

The sum of the terms of a sequence is called a series.

Note 9.40.

Although the words "sequence" and "series" are often used interchangeably in everyday English, they have different meanings in mathematics. A sequence is a list of numbers, whereas a series is a single number obtained by computing a sum. For example, the list of numbers

\begin{equation*} 2, 4, 8, 16, 32 \end{equation*}

is a sequence. The sum of those five terms is the series

\begin{equation*} 2+4+8+16+32=62 \end{equation*}

Checkpoint 9.41. QuickCheck 1.

What is the difference between a sequence and a series?

A) The order of the terms in a series is important, but not in a sequence.
B) A sequence is a list of numbers, and a series is a single number.
C) A series can be used in applications.
D) There is no difference; they are the same.

Answer.

$\text{B) A ... single number.}$

Solution.

A sequence is a list of numbers, and a series is a single number.

We use the symbol $S_n$ to denote the sum of the first $n$ terms of a sequence. Thus, for the sequence in Note 9.40, $S_5 = 62\text{.}$

Example 9.42.

Find the series $S_6$ for the sequence with general term $a_n = 3n + 1\text{.}$

Solution.

The first six terms of the sequence are $4, $ $7, $ $10, $ $13, $ $16, $ and $19. $ Thus,

\begin{equation*} S_6 = 4 + 7 + 10 + 13 + 16 + 19 = 69 \end{equation*}

Checkpoint 9.43. Practice 1.

Find the series $S_5$ for the sequence with general term $b_n = n^2\text{.}$

$S_5=$

Answer.

$55$

Solution.

$1^2+2^2+3^2+4^2+5^2=55$

Subsection Arithmetic Series

We can find a formula for the sum of the first $n$ terms of an arithmetic sequence.

Investigation 9.2. Sum of an Arithmetic Sequence.

Consider the sum of the first 12 terms of the sequence with general term $a_n = 4n+1\text{.}$ That is,

\begin{equation*} S_{12} = 5 + 9 + 13 + \cdots + 41 + 45 + 49 \end{equation*}

In mathematics, as in other fields, making discoveries often depends on looking at familiar objects in a new way.

Let’s write the terms of the series in the opposite order:

\begin{equation*} S_{12} = 49 + 45 + 41 + \cdots + 13 + 9 + 5 \end{equation*}
Now add these two equations term by term, to find

\begin{equation*} \begin{aligned}[t] S_{12} \amp = 5 + 9 + 13 + \cdots + 41 + 45 + 49\\ S_{12} \amp = 49 + 45 + 41 + \cdots + 13 + 9 + 5\\ 2S_{12} \amp = \\ \end{aligned} \end{equation*}
Each term of the sum is the same, namely, 54! (Do you see why?) Also, the term 54 occurs 12 times because we are adding 12 terms of the sequence. We have discoverd that

\begin{equation*} 2S_{12} = 12(54) \end{equation*}

so

\begin{equation*} S_{12} = \dfrac{(\hphantom{0000})(\hphantom{0000})}{(\hphantom{0000})} = \end{equation*}
Notice that the number 54 is the sum of the first term of the arithmetic series, 5, and the last term, 49. In other words, $54 = a_1 + a_n\text{.}$ This is the key observation we need to produce our formula.
In general, this sum $a_1 + a_n$ occurs $n$ times and results in twice the series we want. Thus,

\begin{equation*} 2S_n = a_1 + a_n \end{equation*}

and dividing both sides by 2 gives us the following formula for computing an arithmetic series.

Arithmetic Series.

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is

\begin{equation*} S_n = \dfrac{n}{2} (a_1 + a_n) \end{equation*}

Example 9.44.

Find the sum of the first 15 odd integers.

Solution.

The odd integers form an arithmetic sequence with first term $a_1 = 1$ and common difference 2. Thus, the general term of the sequence is $a_n = 1 + (n - 1)2\text{,}$ and we would like to find the series $S_{15}\text{.}$ We begin by finding the last term of the series, $a_{15}\text{.}$

\begin{equation*} a_{15} = 1 + (15-1)2 = 29 \end{equation*}

Next we use the formula for $S_n$ with $n = 15,~a_1 = 1,$ and $a_{15} = 29\text{:}$

\begin{equation*} S_{15} = \dfrac{15}{2} (1 + 29) = 225 \end{equation*}

Checkpoint 9.45. Practice 2.

Find the sum of the first 100 terms of the sequence $2, 5, 8, 11, \cdots\text{.}$

Answer:

Answer.

$15050$

Solution.

15,050

Example 9.46.

Arlene starts a part-time job in a print shop at a salary of $800 per month. If she keeps up with the training program her salary will increase by $35 per month. How much will Arlene have earned at the end of the 18-month training program?

Solution.

Arlene’s monthly salary is an arithmetic sequence with $a_1 = 800$ and $d = 35\text{.}$ Thus, the general tern of the sequence is $a_n = 800 + (n-1)35\text{,}$ and we would like to find the series $S_{18}\text{.}$ We begin by finding the last term of the series, $a_{18}\text{.}$

\begin{equation*} a_{18} = 800 + (18-1)(35) = 1395 \end{equation*}

Next we use the formula for $S_n$ with $n = 18,~ a_1 = 800\text{,}$ and $a_{18} = 1395\text{.}$

\begin{equation*} S_{18} = \dfrac{18}{2} (800 + 1395) = 19,755 \end{equation*}

Arlene’s total earnings for the 18 months will be $19,755.

Checkpoint 9.47. Practice 3.

Arlene from the previous example decides to start a savings account. After her first paycheck, she deposits $50, and plans to increase her deposits by $5 per month. How much will she have in the savings account at the end of the 18-month training program?

Answer.

$1665$

Solution.

$1665

Subsection Geometric Series

We can also find a formula for the sum of the first $n$ terms of a geometric sequence.

Investigation 9.3. Sum of a Geometric Sequence.

We’ll compute the sum of the first 9 terms of the geometric sequence with $a=3$ and $r=3\text{.}$ Its general term is $a_n = 3(3)^{n-1}\text{,}$ or $3^n\text{.}$ Then

\begin{equation*} S_9 = 3 + 3^2 + 3^3 + \cdots + 3^9 \end{equation*}

Instead of writing the terms in the opposite order, as we did for the arithmetic series, this time we will multiply each term of $S_9$ by the common ratio, 3.

\begin{equation*} 3 \cdot S_9 = 3 \cdot 3 + 3 \cdot 3^2 + 3 \cdot 3^3 + \cdots + 3 \cdot 3^8 + 3 \cdot 3^9 \end{equation*}

or

\begin{equation*} 3S_9 = 3^2 + 3^3 + 3^4 + \cdots + 3^9 + 3^{10} \end{equation*}
Next we subtract the first equation from the second equation. Most of the terms will cancel out in the subtraction.

\begin{equation*} \begin{aligned}[t] 3S_9 \amp = ~~~~~~~~~~~3^2 + 3^3 + \cdots + 3^9 + 3^{10}\\ -[S_9 \amp = ~~~3 + 3^2 + 3^3 + \cdots + 3^9]\\ 2S_9 \amp = -3 ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + 3^{10}\\ \end{aligned} \end{equation*}
Thus

\begin{equation*} 2S_9 = 3^{10} - 3 \end{equation*}

so

\begin{equation*} S_9 = \dfrac{(\hphantom{0000}) - (\hphantom{0000})}{(\hphantom{0000})} = \end{equation*}
Notice that the second term of the numerator is $a_1$ (3 in this example) and the first term is $a_{10}~ (3 \cdot 3^9$ or $3^{10})\text{.}$
Although you might not guess it from this example, the denominator of the expression, 2, is equal to $r-1\text{.}$ In general, we can derive the following formula for computing a geometric series.

Geometric Series.

The sum $S_n$ of the first $n$ terms of a geometric sequence is

\begin{equation*} S_n = \dfrac{a_{n+1} - a_1}{r-1} \end{equation*}

Checkpoint 9.48. QuickCheck 2.

To find the series for a given sequence, we

Compute the last term of the sequence.
Find the average value of the terms.
Add the terms.
List the terms in the opposite order.

Answer.

$\text{Add the terms.}$

Solution.

Add the terms.

Example 9.49.

Find the sum of the first five terms of the sequence $5, \dfrac{10}{3}, \dfrac{20}{9}, \dfrac{40}{27}, \cdots\text{.}$

Solution.

The sequence is geometric with $a_1 = 5\text{.}$ We find by dividing $\dfrac{10}{3}$ by 5.

\begin{equation*} \dfrac{10}{3} \div 5 = \dfrac{2}{3} \end{equation*}

Thus, the general term of the series is $a_n = 5\left(\dfrac{2}{3}\right)^{n-1}\text{.}$ We use the formula for a geometric series with $n=5\text{,}$

\begin{equation*} S_5 = \dfrac{a_6 - a_1}{r-1} \end{equation*}

We substitute $a_1 = 5,~a_6 = 5\left(\dfrac{2}{3}\right)^5\text{,}$ and $r=\dfrac{2}{3}\text{.}$ Thus

\begin{equation*} \begin{aligned}[t] S_5 \amp = \dfrac{5\left(\dfrac{2}{3}\right)^5 - 5}{\dfrac{2}{3}-1} = \dfrac{5\left(\dfrac{32}{243} - 1\right)}{-\dfrac{1}{3}}\\ \amp = 5\left(\dfrac{32-243}{243}\right)\left(\dfrac{-3}{1}\right) \approx 13.02\\ \end{aligned} \end{equation*}

Checkpoint 9.50. Practice 4.

Find the sum of the first ten terms of the geometric sequence $100\text{,}$ $110\text{,}$ $121,~ \ldots\text{.}$

Answer.

$1593.74$

Solution.

1593.74

Example 9.51.

Payam’s starting salary as an engineer is $60,000 with a 5% annual raise for each of the next five years, depending on suitable progress. If Payam receives each salary increase, how much will he make over the next six years?

Solution.

Payam’s salary is multiplied each year by a factor of 1.05, so its values form a geometric sequence with a common ratio of $r=1.05\text{.}$ The general term of the sequence is $a_n = 60,000(1.05)^{n-1}\text{.}$ His total income over the six years will be the sum of the first six terms of the sequence. Thus,

\begin{equation*} \begin{aligned}[t] S_6 \amp = \dfrac{a_7 - a_1}{r-1} = \dfrac{60,000(1.05)^6 - 60,000}{1.05-1}\\ \amp = 408,114.77 \\ \end{aligned} \end{equation*}

Payam will earn $408,114.77 over the next six years.

Checkpoint 9.52. Practice 5.

Show that the formula for a geometric series can also be written as

\begin{equation*} \begin{gathered} S_n = \dfrac{a_1(r^n - 1)}{r-1} \end{gathered} \end{equation*}

Check all the relevant statements.

The sum $S_n$ of the first $n$ terms of a geometric series is
- $\displaystyle S_n = \dfrac{a_{n+1} - a_1}{r-1}$
- $\displaystyle S_n = \dfrac{n}{2}(a_{1} + a_n)$
- $\displaystyle S_n = a_1 r^n $
- $\displaystyle S_n = a + (n-1)r $
We can write $a_{n+1}$ as
- $\displaystyle a_{n+1} = a_{1} + nr $
- $\displaystyle a_{n+1} = a_{1} + (n-1)r $
- $\displaystyle a_{n+1} = a_1 r^n $
- $\displaystyle a_{n+1} = a_1 r^{n-1} $
So the numerator from part (a) can be rewritten as
- $\displaystyle a_{1} + nr -a_1 = a_{1} (r^n-1) $
- $\displaystyle a_{1} + (n-1)r - a_1 = a_{1} (r^n-1) $
- $\displaystyle a_1 r^n -a_1 = a_{1} (r^n-1) $
- $\displaystyle a_1 r^{n-1} - a_1 = a_{1} (r^n-1) $

Answer 1.

$\text{Choice 1}$

Answer 2.

$\text{Choice 3}$

Answer 3.

$\text{Choice 3}$

Solution.

$S_n = \dfrac{a_{n+1} - a_1}{r-1} = \dfrac{a_1r^n - a_1}{r-1} = \dfrac{a_1(r^n - 1)}{r-1}$

Subsection Section Summary

Subsubsection Vocabulary

Series
Arithmetic series
Geometric series

Subsubsection CONCEPTS

The sum of the terms of a sequence is called a series.
The sum $S_n$ of the first $n$ terms of an arithmetic sequence is

\begin{equation*} S_n = \dfrac{n}{2} (a_1 + a_n) \end{equation*}
The sum $S_n$ of the first $n$ terms of a geometric sequence is

\begin{equation*} S_n = \dfrac{a_{n+1} - a_1}{r-1} \end{equation*}

Subsubsection STUDY QUESTIONS

What is the difference between a sequence and a series?
Describe a method for finding the sum of an arithmetic sequence.
Describe a method for finding the sum of a geometric sequence.
How can you decide whether a given sequence is arithmetic or geometric (or neither)?

Subsubsection SKILLS

Practice each skill in the Homework Problems listed.

Evaluate arithmetic and geometric series: #1-12
Identify a series as arithmetic or geometric: #13-22
Model a situation with a series: #23-40

Exercises Homework 9.3

Exercise Group.

For Problems 1–6, evaluate the arithmetic series.

1.

The sum of the first nine terms of the sequence $~a_n = -4 + 3n$

2.

The sum of the first ten terms of the sequence $~a_n = 5-2n$

3.

The sum of the first 16 terms of the sequence $~a_n = 18-\dfrac{4}{3}n$

4.

The sum of the first 13 terms of the sequence $~a_n = -6 - \dfrac{1}{2}n$

5.

The sum of the first 30 terms of the sequence $~a_n = 1.6+0.2n$

6.

The sum of the first 25 terms of the sequence $~a_n = 2.5+0.3n$

Exercise Group.

For Problems 7–12, evaluate the geometric series.

7.

The sum of the first five terms of $a_n = 2(-4)^{n-1}$

8.

The sum of the first eight terms of $a_n = 12(3)^{n-1}$

9.

The sum of the first nine terms of $a_n = -48\left(\dfrac{1}{2}\right)^{n-1}$

10.

The sum of the first six terms of $a_n = 81\left(\dfrac{2}{3}\right)^{n-1}$

11.

The sum of the first four terms of $a_n = 18(1.15)^{n-1}$

12.

The sum of the first four terms of $a_n = 512(0.72)^{n-1}$

Exercise Group.

For Problems 13–22, identify the series as arithmetic, geometric or neither, then evaluate it.

13.

$2+4+6+\cdots+96+98+100$

14.

$1+3+5+\cdots+95+97+99$

15.

$2+4+8+16+\cdots+256+512+1024$

16.

$1+3+9+27+\cdots+6561+19,683$

17.

$1+8+24+64+125+216+343$

18.

$1+11+111+1111+11,111+111,111$

19.

$87+84+81+78+\cdots+45+42+39$

20.

$1+(-2)+(-5)+\cdots+(-41)+(-44)$

21.

$6+2+\dfrac{2}{3}+\cdots+\dfrac{2}{81}+\dfrac{2}{243}$

22.

$12+3+\dfrac{3}{4}+\cdots+\dfrac{3}{64}+\dfrac{3}{128}$

Exercise Group.

For Problems 23–40, write a series to describe the problem, then evaluate it.

23.

Find the sum of all the even integers from 14 to 88.

24.

Find the sum of all multiples of 7 from 14 to 105.

25.

A clock strikes once at one o’clock, twice at two o’clock, and so on. How many times will the clock strike in a twelve hour period?

26.

Jessica puts one candle on the cake at her daughter’s first birthday, two candles at her second birthday, and so on. How many candles will Jessica have used after her daughter’s sixteenth birthday?

27.

A rubber ball is dropped from a height of 24 feet and returns to three-fourths of its previous height on each bounce.

How high does the ball bounce after hitting the floor for the third time?
How far has the ball traveled vertically when it hits the floor for the fourth time?

28.

A Yorkshire terrier can jump 3 feet into the air on his first bounce and five-sixths the height of his previous jump on each successive bounce.

How high can the terrier go on his fourth bounce?
How far has the terrier traveled vertically when he returns to the ground after his fourth bounce?

29.

Sales of Brussels Sprouts dolls peaked at $920,000 in 1991 and began to decline at a steady rate of $40,000 per year. What total revenue did the manufacturer gain from sale of the dolls from 1991 to 2000?

30.

It takes Alida 20 minutes to type the first page of her term paper, but each subsequent page takes her 40 seconds less than the previous one. How long will it take her to type her 30-page paper?

31.

A computer takes 0.1 second to perform the first iteration of a certain loop, and each subsequent iteration takes 0.05 seconds longer than the previous one. How long will it take the computer to perform 50 iterations?

32.

Richard’s water bill was $63.50 last month. If his bill increases by $2.30 per month, how much should he expect to pay for water during the next 10 months?

33.

Sales of Energy Ranger dolls peaked at $920,000 in 2001 and began to decline by 8% per year. What total revenue did the manufacturer gain from sale of the dolls from 2001 to 2010?

34.

It takes Emily 20 minutes to type the first page of her term paper, but each subsequent page takes only 95% as long as the previous one. How long will it take her to type her 30-page paper?

35.

A computer takes 0.1 second to perform the first iteration of a certain loop, and each subsequent iteration requires 20% longer than the previous one. How long will it take the computer to perform 50 iterations?

36.

Megan’s water bill was $63.50 last month. If her bill increases by 2% per month, how much should she expect to pay for water during the next 10 months?

37.

Jim and Nora begin a college fund for their son David by depositing $500 into an account each year, beginning on the day David was born. If the account earns an interest rate of 5% compounded annually, how much will be in the account on David’s eighteenth birthday?

38.

Ben begins an Individual Retirement Account when he turns 25 years old, depositing $2000 into the account each year. If the account earns 6% interest compounded annually, how much will he have in the account when he turns 65 years old?

39.

Suppose that you are given 1¢ on the first day of the month, 2¢ on the second day, 4¢ on the third day, and so on, each day’s payment being twice the previous day’s. What would be your total income on the thirtieth day?

40.

According to legend, a man who had pleased the Persian king asked for the following reward. The man was to receive a single grain of wheat for the first square of a chessboard, two grains for the second square, four grains for the third square, and so on, doubling the amount for each square up to the sixty-fourth square. How many grains would he receive in all?

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